PDwei

PDwei computes weight function psi(u)/u for for minimum density power divergence estimator

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Syntax

Description

example

w =PDwei(u, alpha) Weight function for minimum density power divergence estimator.

Examples

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  • Weight function for minimum density power divergence estimator.

    • x=-6:0.01:6;
alpha=0.01;
weiPD=PDwei(x,alpha);
plot(x,weiPD)
xlabel('x','Interpreter','Latex')
ylabel('$W (x) =\psi(x)/x$','Interpreter','Latex')

    Related Examples

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  • Comparing two weight functions for two values of bdp.

    • close all
x=-6:0.01:6;
lwd=2;
hold('on')
bdp1=0.25;
alpha1=PDbdp(bdp1);    
weiPD=PDwei(x,alpha1);
weiPD=weiPD/max(weiPD);
plot(x,weiPD,'LineStyle','-','LineWidth',lwd)
bdp2=0.01;
c=PDbdp(bdp2);
weiPD=PDwei(x,c);
weiPD=weiPD/max(weiPD);
plot(x,weiPD,'LineStyle','-.','LineWidth',lwd)
xlabel('$x$','Interpreter','Latex','FontSize',16)
ylabel('PD weight function $\psi_\alpha(x)/x$','Interpreter','Latex','FontSize',20)
legend({['bdp=' num2str(bdp1,2)],  ['bdp=' num2str(bdp2,2)]},...
'Interpreter','Latex','Location','SouthEast','FontSize',12)

  • Compare five different weight functions.

    • In each of them eff is 95 per cent. 

    % Initialize graphical parameters.
FontSize=14;
x=-6:0.01:6;
ylim1=-0.05;
ylim2=1.05;
xlim1=min(x);
xlim2=max(x);
LineWidth=2;
subplot(2,3,1)
ceff095HU=HUeff(0.95,1);
weiHU=HUwei(x,ceff095HU);
plot(x,weiHU,'LineWidth',LineWidth)
xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
title('Huber','FontSize',FontSize)
ylim([ylim1 ylim2])
xlim([xlim1 xlim2])
subplot(2,3,2)
ceff095HA=HAeff(0.95,1);
weiHA=HAwei(x,ceff095HA);
plot(x,weiHA,'LineWidth',LineWidth)
xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
title('Hampel','FontSize',FontSize)
ylim([ylim1 ylim2])
xlim([xlim1 xlim2])
subplot(2,3,3)
ceff095TB=TBeff(0.95,1);
weiTB=TBwei(x,ceff095TB);
plot(x,weiTB,'LineWidth',LineWidth)
xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
title('Tukey biweight','FontSize',FontSize)
ylim([ylim1 ylim2])
xlim([xlim1 xlim2])
subplot(2,3,4)
ceff095HYP=HYPeff(0.95,1);
ktuning=4.5;
weiHYP=HYPwei(x,[ceff095HYP,ktuning]);
plot(x,weiHYP,'LineWidth',LineWidth)
xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
title('Hyperbolic','FontSize',FontSize)
ylim([ylim1 ylim2])
xlim([xlim1 xlim2])
subplot(2,3,5)
ceff095PD=PDeff(0.95);
weiPD=PDwei(x,ceff095PD);
weiPD=weiPD/max(weiPD);
plot(x,weiPD,'LineWidth',LineWidth)
xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
title('Power divergence','FontSize',FontSize)
ylim([ylim1 ylim2])
xlim([xlim1 xlim2])
    Click here for the graphical output of this example (link to Ro.S.A. website)

    Input Arguments

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    u — scaled residuals or Mahalanobis distances. Vector.

    n x 1 vector containing residuals or Mahalanobis distances for the n units of the sample

    Data Types: single| double

    alpha — tuning parameter. Scalar.

    Scalar in the interval (0,1] which controls the robustness/efficiency of the estimator (beta in regression or mu in the location case ...). The greater is alpha the greater is the bdp and smaller is the efficiency.

    Data Types: single| double

    Output Arguments

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    w —Tukey's biweight weights associated to the scaled residuals or Mahalanobis distances for the n units of the sample. n -by- 1 vector

    More About

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    Additional Details

    function PDwei transforms vector u as follows \[ PDwei(u,alpha)= \alpha \exp(-\alpha (u^2/2)); \]

    References

    Riani, M. Atkinson, A.C., Corbellini A. and Perrotta A. (2020), Robust Regression with Density Power Divergence: Theory, Comparisons and Data Analysis, Entropy, Vol. 22, 399.

    https://www.mdpi.com/1099-4300/22/4/399

    See Also

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